Stochastic differential equations (SDEs) are flexible models for describing system behaviors from various contexts. We propose a way to adopt the Gaussian process regression method to approximate the conditional probability distribution of the state vector whose evolution is described by SDE with linear drift. We mathematically derive the exact covariance structure of a multi-dimensional diffusion process with linear drift, and use it as a kernel function in multi-output Gaussian process regression. The usage of our approach differs according to the knowledge regarding the parameters in SDE. For the case when the coefficients of the SDEs are known, we firstly compare our approach with sampling-based methodologies for estimating the conditional probability distribution of diffusion processes by taking Cox-Ingersoll-Ross model as an example. The results show that our approach is consistent with sampling algorithms considering the estimated mean and variance. We then illustrate how to adopt our approach when the coefficients of the SDEs are unknown. We compare the performance of Gaussian process regression with the kernel function we derived to those with conventional kernel candidates. When applied to multi-dimensional Ornstein-Uhlenbeck process and inlet/outlet pressure data from gas regulators, our method considerably reduces the mean square error of prediction tasks, especially when there is information that comes across dimensions.
본 논문에서는 선형적 추세를 가지는 확률 미분 방정식에 대한 정규 과정 회귀 기법에 대해 다룬다. 선형 추세 확률 미분 방정식의 시공간에 대한 공분산을 유도하고, 이를 다차원 정규 과정 회귀법의 매개변수인 공분산 함수로 활용하는 방안을 제시한다. 회귀의 목적이 되는 확률 미분 방정식의 계수의 정보 유무에 따라 시뮬레이션을 통해 다차원 확산 과정의 조건부 확률 분포를 근사하거나, 기울기 기반의 최적화 알고리즘을 활용하여 계수를 추정하고 이를 다차원 정규 과정 회귀의 관점에서 활용하는 과정을 제안한다.
